Cobar construction

Cobar construction
Name	Cobar construction
Type	Differential graded algebra
Introduced	1960s
Authors	Jean-Louis Loday; development by Gerhard Hochschild; formalized in work of Daniel Quillen and Jean-Pierre Serre

Cobar construction The Cobar construction is a standard machine in homological algebra producing a differential graded algebra from a differential graded coalgebra, with deep connections to the Bar construction, Koszul duality, and models in rational homotopy theory. It appears in the work of figures such as Daniel Quillen, Jean-Pierre Serre, Jean-Louis Loday, and Jean-Michel Bismut and plays a role in computations linked to Eilenberg–MacLane space, Dennis Sullivan models, and Adams spectral sequence analyses.

Definition and basic construction

Given a differential graded coalgebra C over a field k (often connected or conilpotent), the Cobar construction is the tensor algebra T(s^{-1}\overline{C}) on the desuspension s^{-1}\overline{C} of the coaugmentation coideal \overline{C}, endowed with a differential determined by the internal differential of C and the reduced coproduct. Historically formulated in the context of work by Samuel Eilenberg and Saunders Mac Lane and refined in Daniel Quillen’s homotopical algebra approach, the construction yields a differential graded algebra A = ΩC whose underlying graded algebra is free as a tensor algebra on the graded vector space s^{-1}\overline{C}. For conilpotent coalgebras arising from chains on a pointed based space, this procedure connects to classical models used by J. Peter May and J. H. C. Whitehead.

Algebraic properties and structure

The Cobar construction ΩC inherits a grading and a differential making it a differential graded algebra (DGA) that is augmented, often yielding an augmented model comparable to the universal enveloping algebra in the Lie setting studied by Jean-Pierre Serre and Murray Gerstenhaber. When C is coaugmented and conilpotent, ΩC is quasi-free and exhibits homological properties consistent with results of Gerstenhaber and Martin Markl on algebraic deformation theory. The homology H_*(ΩC) carries algebra structures that reflect coalgebraic primitives in C, echoing phenomena studied in the context of Hopf algebras by Nicholas Jacobson and Milnor–Moore theorem-type results considered by John Milnor and John C. Moore. Functoriality under morphisms of coalgebras leads to Quillen equivalences in the model category frameworks developed by Daniel Quillen and employed by Vladimir Hinich.

Relationships with bar construction and Koszul duality

The Cobar and Bar constructions form an adjoint pair (Ω, B) relating augmented DGAs and coaugmented DGCs, an adjunction exploited in Quillen’s work on rational homotopy and in dualities articulated by Jean-Louis Loday and Bruno Vallette. For Koszul algebras studied by Roger Berger and S. Priddy, the Koszul duality formalism identifies Ω of the Koszul dual coalgebra with resolutions used by Henri Cartan and Samuel Eilenberg in homological calculations. In the operadic context, the interplay of Bar and Cobar underpins descriptions by Vladimir Ginzburg and M. Markl of homotopy algebras such as A∞-algebras and L∞-algebras, and is central to formalisms introduced by Gérard Laumon and Maxim Kontsevich.

Examples and computations

Concrete computations include: ΩC for C the chains on an interval yields a model quasi-isomorphic to the tensor algebra on a single generator with trivial differential, echoing early calculations by Eilenberg–Mac Lane and J. F. Adams; when C is the reduced chains on an odd-dimensional sphere, ΩC produces an algebra reflecting the loop space homology calculations of Jean-Pierre Serre and Raoul Bott; for cocommutative coalgebras arising from universal enveloping coalgebras of Lie algebras, Ω recovers constructions related to the Poincaré–Birkhoff–Witt theorem studied by Élie Cartan and Anatoly Shirshov. Explicit spectral sequence computations invoking the Adams spectral sequence or the Bousfield–Kan spectral sequence are standard in work by Douglas Ravenel and Samuel Gitler.

Applications in homological algebra and rational homotopy

In rational homotopy theory, Ω of the coalgebra of singular chains yields models for based loop spaces used by Dennis Sullivan and Daniel Quillen to compare minimal models and Lie models; this underlies computations in the study of nilpotent spaces considered by John Milnor and John C. Moore. The Cobar construction provides algebraic resolutions employed in computing Ext and Tor groups in the style of Cartan–Eilenberg and appears in deformation problems treated by Martin Kontsevich and Edward Witten in topological field theory contexts. In the representation-theoretic setting, Ω of coalgebras associated to finite-dimensional algebras relates to derived categories explored by Bernhard Keller and Joseph Bernstein.

Variations and generalizations

Generalizations include Cobar constructions parameterized by operads, leading to Ω_P for an operad P as developed by Victor Ginzburg and Bruno Vallette, and homotopy Cobar constructions realizing A∞ and L∞ structures studied by Jim Stasheff and Tomasz Kadeishvili. Relative and completed Cobar constructions appear in profinite and p-adic contexts in work by Jean-Pierre Serre and John Tate, while equivariant versions interact with group cohomology theories developed by Roger Lyndon and Kenneth Brown. Higher categorical and infinity-categorical refinements connect to frameworks pioneered by Jacob Lurie and André Joyal.

Category:Homological algebra